JEE Main & Advanced Mathematics Applications of Derivatives Question Bank

done Tangent and Normal Total Questions - 39

• question_answer1)
  The points on the curve \[y=12x-{{x}^{3}}\]at which the gradient is zero are [MP PET 1999]
  A)
             (0, 2), (2,16) done clear
  B)
             (0, ? 2), (2, ? 16) done clear
  C)
             (2, ?16), (? 2, 16) done clear
  D)
             (2, 16), (? 2, ? 16) done clear
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• question_answer2)
  The area of the triangle formed by the coordinate axes and a tangent to the curve \[xy={{a}^{2}}\]at the point \[({{x}_{1}},{{y}_{1}})\]on it is                      [DCE 2001]
  A)
             \[\frac{{{a}^{2}}{{x}_{1}}}{{{y}_{1}}}\] done clear
  B)
             \[\frac{{{a}^{2}}{{y}_{1}}}{{{x}_{1}}}\] done clear
  C)
             \[2{{a}^{2}}\] done clear
  D)
             \[4{{a}^{2}}\] done clear
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• question_answer3)
  The slope of tangent to the curve\[x={{t}^{2}}+3t-8\], \[y=2{{t}^{2}}-2t-5\] at the point (2, ?1) is [MNR 1994]
  A)
             \[\frac{22}{7}\] done clear
  B)
             \[\frac{6}{7}\] done clear
  C)
             ? 6  done clear
  D)
             None of these done clear
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• question_answer4)
  The point of the curve \[{{y}^{2}}=2(x-3)\] at which the normal is parallel to the line\[y-2x+1=0\]is           [MP PET 1998]
  A)
              (5,2) done clear
  B)
             \[\left( -\frac{1}{2},-2 \right)\] done clear
  C)
             (5, ?2) done clear
  D)
             \[\left( \frac{3}{2},\,2 \right)\] done clear
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• question_answer5)
  The line \[x+y=2\]is tangent to the curve \[{{x}^{2}}=3-2y\] at its point [MP PET 1998]
  A)
             (1, 1) done clear
  B)
             (?1, 1) done clear
  C)
             (\[\sqrt{3}\], 0) done clear
  D)
             (3, ?3) done clear
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• question_answer6)
  If \[x={{t}^{2}}\]and \[y=2t\], then equation of the normal at \[t=1\]is                       [RPET 1996]
  A)
             \[x+y-3=0\] done clear
  B)
             \[x+y-1=0\] done clear
  C)
             \[x+y+1=0\] done clear
  D)
             \[x+y+3=0\] done clear
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• question_answer7)
  The equation of the normal to the curve \[y=\sin \frac{\pi x}{2}\]at (1, 1) is                          [AMU 1999]
  A)
             \[y=1\] done clear
  B)
             \[x=1\] done clear
  C)
             \[y=x\] done clear
  D)
             \[y-1=\frac{-2}{\pi }(x-1)\] done clear
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• question_answer8)
  The equation of tangent to the curve \[y=2\cos x\] at \[x=\frac{\pi }{4}\] is                                                         [RPET 1997]                                          
  A)
             \[y-\sqrt{2}=2\sqrt{2}\left( x-\frac{\pi }{4} \right)\] done clear
  B)
             \[y+\sqrt{2}=\sqrt{2}\left( x+\frac{\pi }{4} \right)\] done clear
  C)
             \[y-\sqrt{2}=-\sqrt{2}\left( x-\frac{\pi }{4} \right)\] done clear
  D)
             \[y-\sqrt{2}=\sqrt{2}\left( x-\frac{\pi }{4} \right)\] done clear
  View Solution play_arrow

• question_answer9)
  At which point the line \[\frac{x}{a}+\frac{y}{b}=1\], touches the curve \[y=b{{e}^{-x/a}}\]         [RPET 1999]
  A)
             (0, 0) done clear
  B)
             (0, a) done clear
  C)
             (0, b) done clear
  D)
             (b, 0) done clear
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• question_answer10)
  The angle between curves \[{{y}^{2}}=4x\] and \[{{x}^{2}}+{{y}^{2}}=5\]at              (1, 2)  is                                             [Karnataka CET 1999]
  A)
             \[{{\tan }^{-1}}(3)\] done clear
  B)
             \[{{\tan }^{-1}}(2)\] done clear
  C)
             \[\frac{\pi }{2}\] done clear
  D)
             \[\frac{\pi }{4}\] done clear
  View Solution play_arrow

• question_answer11)
  For the curve \[b{{y}^{2}}={{(x+a)}^{3}}\]the square of subtangent is proportional to     [RPET 1999]
  A)
             \[{{\text{(Subnormal)}}^{1/2}}\] done clear
  B)
             Subnormal done clear
  C)
             \[{{\text{(Subnormal)}}^{\text{3/2}}}\] done clear
  D)
             None of these done clear
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• question_answer12)
  The tangent to the curve \[y=a{{x}^{2}}+bx\] at \[(2,\,-8)\] is parallel to x-axis. Then       [AMU 1999]
  A)
             \[a=2,\,b=-2\] done clear
  B)
             \[a=2,\,\,b=-4\] done clear
  C)
             \[a=2\,\,b=-8\] done clear
  D)
             \[a=4,\,b=-4\] done clear
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• question_answer13)
  The sum of intercepts on co-ordinate axes made by tangent to the curve \[\sqrt{x}+\sqrt{y}=\sqrt{a}\]is                                [RPET 1999]
  A)
             \[a\] done clear
  B)
             \[2a\] done clear
  C)
             \[2\sqrt{a}\] done clear
  D)
             None of these done clear
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• question_answer14)
  Co-ordinates of a point on the curve \[y=x\log x\] at which the normal is parallel to the line \[2x-2y=3\] are                    [RPET 2000]
  A)
             (0,0) done clear
  B)
             \[(e,\,\,e)\] done clear
  C)
             \[({{e}^{2}},\,2{{e}^{2}})\] done clear
  D)
             \[({{e}^{-2}}-2{{e}^{-2}})\] done clear
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• question_answer15)
  If normal to the curve \[y=f(x)\] is parallel to x-axis, then correct statement is                                    [RPET 2000]
  A)
             \[\frac{dy}{dx}=0\] done clear
  B)
             \[\frac{dy}{dx}=1\] done clear
  C)
             \[\frac{dx}{dy}=0\] done clear
  D)
             None of these done clear
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• question_answer16)
  The length of normal to the curve \[x=a\,(\theta +\sin \theta ),\] \[y=a(1-\cos \theta )\] at the point \[\theta =\pi /2\]is            [RPET 1999]
  A)
             \[2a\] done clear
  B)
             \[a/2\] done clear
  C)
             \[\sqrt{2}\,a\] done clear
  D)
             \[a/\sqrt{2}\] done clear
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• question_answer17)
  The normal of the curve \[x=a(\cos \theta +\theta \sin \theta )\] \[y=a(\sin \theta -\theta \cos \theta )\] at any \[\theta \] is such that            [DCE 2000; AIEEE 2005]
  A)
             It makes a constant angle with x-axis done clear
  B)
             It passes through the origin done clear
  C)
             It is at a constant distance from the origin done clear
  D)
             None of these done clear
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• question_answer18)
  The slope of the tangent to the curve \[x=3{{t}^{2}}+1,y={{t}^{3}}-1\] at \[x=1\] is                                           [Karnataka CET 2003]
  A)
             0 done clear
  B)
             \[\frac{1}{2}\] done clear
  C)
             \[\infty \] done clear
  D)
             \[-2\] done clear
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• question_answer19)
  An equation of the tangent to the curve \[y={{x}^{4}}\] from the point (2, 0) not on the curve is                                [RPET 2000]
  A)
             \[y=0\] done clear
  B)
             \[x=0\] done clear
  C)
             \[x+y=0\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer20)
  The angle of intersection of the curves \[y={{x}^{2}}\] and \[x={{y}^{2}}\] at (1, 1) is [Roorkee 2000; Karnataka CET 2001]
  A)
             \[{{\tan }^{-1}}\left( \frac{4}{3} \right)\] done clear
  B)
             \[{{\tan }^{-1}}(1)\] done clear
  C)
             \[{{90}^{o}}\] done clear
  D)
             \[{{\tan }^{-1}}\left( \frac{3}{4} \right)\] done clear
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• question_answer21)
  The abscissae of the points, where the tangent to curve \[y={{x}^{3}}-3{{x}^{2}}-9x+5\]is parallel to x-axis, are                    [Karnataka CET 2001]
  A)
             0 and 0 done clear
  B)
             x = 1 and ? 1 done clear
  C)
             x = 1 and ? 3 done clear
  D)
             x =  ? 1 and 3 done clear
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• question_answer22)
  If the curve \[y={{a}^{x}}\] and \[y={{b}^{x}}\] intersect at angle \[\alpha \] then,\[\tan \alpha =\]                                     [MP PET 2001]
  A)
             \[\frac{a-b}{1+ab}\] done clear
  B)
             \[\frac{\log a-\log b}{1+\log a\log b}\] done clear
  C)
             \[\frac{a+b}{1-ab}\] done clear
  D)
             \[\frac{\log a+\log b}{1-\log a\log b}\] done clear
  View Solution play_arrow

• question_answer23)
  The equation of tangent at \[(-4,\,-4)\] on the curve \[{{x}^{2}}=-4y\] is [Karnataka CET 2001: Pb. CET 2000]
  A)
             \[2x+y+4=0\] done clear
  B)
             \[2x-y-12=0\] done clear
  C)
             \[2x+y-4=0\] done clear
  D)
             \[2x-y+4=0\] done clear
  View Solution play_arrow

• question_answer24)
  The point at which the tangent to the curve \[y=2{{x}^{2}}-x+1\] is parallel to \[y\text{ }=\text{ 3}x+\text{9 }\]will be                      [Karnataka CET 2001]
  A)
             (2, 1) done clear
  B)
             (1, 2) done clear
  C)
             (3, 9) done clear
  D)
             (?2, 1) done clear
  View Solution play_arrow

• question_answer25)
  At what point on the curve \[{{x}^{3}}-8{{a}^{2}}y=0\], the slope of the normal is \[\frac{-2}{3}\] [RPET 2002]
  A)
             \[(a,\,a)\] done clear
  B)
             \[(2a,\,-a)\] done clear
  C)
             \[(2a,\,a)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer26)
  The length of the normal at point ?t? of the curve \[x=a(t+\sin t),\] \[y=a(1-\cos t)\] is [RPET 2001]
  A)
             \[a\sin t\] done clear
  B)
             \[2a{{\sin }^{3}}(t/2)\sec (t/2)\] done clear
  C)
             \[2a\sin (t/2)\,\,\tan \,(t/2)\] done clear
  D)
             \[2a\sin (t/2)\] done clear
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• question_answer27)
  The tangent drawn at the point (0, 1) on the curve \[y={{e}^{2x}}\] meets x-axis at the point                                      [RPET 2002]
  A)
             \[(1/2,\,0)\] done clear
  B)
             \[(-1/2,\,\,0)\] done clear
  C)
             (2, 0) done clear
  D)
             \[(0,\,\,0)\] done clear
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• question_answer28)
  The equation of the tangent to the curve \[(1+{{x}^{2}})y=2-x,\] where it crosses the x-axis, is                                    [Kerala (Engg.) 2002]
  A)
             \[x+5y=2\] done clear
  B)
             \[x-5y=2\] done clear
  C)
             \[5x-y=2\] done clear
  D)
             \[5x+y-2=0\] done clear
  View Solution play_arrow

• question_answer29)
  The equation of the tangent to curve \[y=b{{e}^{-x/a}}\] at the point where it crosses y-axis is                             [Karnataka CET 2002]
  A)
              \[ax+by=1\] done clear
  B)
              \[ax-by=1\] done clear
  C)
             \[\frac{x}{a}-\frac{y}{b}=1\] done clear
  D)
             \[\frac{x}{a}+\frac{y}{b}=1\] done clear
  View Solution play_arrow

• question_answer30)
  The angle of intersection of curves \[y={{x}^{2}},\] \[6y=7-{{x}^{3}}\] at (1, 1) is [Kurukshetra CEE 2002]
  A)
             \[\pi /4\] done clear
  B)
             \[\pi /3\] done clear
  C)
             \[\pi /2\] done clear
  D)
             \[\pi \] done clear
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• question_answer31)
  The tangent to the curve \[y=2{{x}^{2}}-x+1\] at a point P is parallel to \[y=3x+4,\]the co-ordinates of P are                                                               [RPET 2003]
  A)
             (2, 1) done clear
  B)
             (1, 2) done clear
  C)
             (? 1, 2) done clear
  D)
             (2, ? 1) done clear
  View Solution play_arrow

• question_answer32)
  For the curve \[xy={{c}^{2}}\] the subnormal at any point varies as                                                               [Karnataka CET 2003]
  A)
             \[{{x}^{2}}\] done clear
  B)
             \[{{x}^{3}}\] done clear
  C)
             \[{{y}^{2}}\] done clear
  D)
             \[{{y}^{3}}\] done clear
  View Solution play_arrow

• question_answer33)
  The angle between the curves \[y=\sin x\] and \[y=\cos x\] is                                                               [EAMCET 2003]
  A)
             \[{{\tan }^{-1}}(2\sqrt{2})\] done clear
  B)
             \[{{\tan }^{-1}}(3\sqrt{2})\] done clear
  C)
             \[{{\tan }^{-1}}(3\sqrt{3})\] done clear
  D)
             \[{{\tan }^{-1}}(5\sqrt{2})\] done clear
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• question_answer34)
   If the normal to the curve \[{{y}^{2}}=5x-1\], at the point (1, ?2) is of the form \[ax-5y+b=0\], then a and b are                                                               [Pb. CET 2001]
  A)
             4, ? 14 done clear
  B)
             4, 14 done clear
  C)
             ?4, 14 done clear
  D)
             ?4, ?14 done clear
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• question_answer35)
  If a tangent to the curve \[y=6x-{{x}^{2}}\]is parallel to the line \[4x-2y-1=0\], then the point of tangency on the curve is                                                               [Karnataka CET 2004]
  A)
             (2, 8) done clear
  B)
             (8, 2) done clear
  C)
             (6, 1) done clear
  D)
             (4, 2) done clear
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• question_answer36)
  The normal to the curve \[x=a\text{ }(1+\cos \theta ),\,y=a\sin \theta \]at \['\theta '\] always passes through the fixed point                                        [AIEEE 2004]
  A)
             (a, a) done clear
  B)
             (0, a) done clear
  C)
             (0, 0) done clear
  D)
             (a, 0) done clear
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• question_answer37)
  . If ST and SN are the lengths of the subtangent and the subnormal at the point \[\theta =\frac{\pi }{2}\]on the curve \[x=a(\theta +\sin \theta ),y=a(1-\cos \theta ),a\ne 1\], then            [Karnataka CET  2005]
  A)
             \[ST=SN\] done clear
  B)
             \[ST=2\,SN\] done clear
  C)
             \[S{{T}^{2}}=a\,S{{N}^{3}}\] done clear
  D)
             \[S{{T}^{3}}=a\,SN\] done clear
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• question_answer38)
  The equation of the tangent to the curve \[x=2{{\cos }^{3}}\theta \] and \[y=3{{\sin }^{3}}\theta \] at the point \[\theta =\pi /4\] is                                         [J & K 2005]
  A)
             \[2x+3y=3\sqrt{2}\] done clear
  B)
             \[2x-3y=3\sqrt{2}\] done clear
  C)
             \[3x+2y=3\sqrt{2}\] done clear
  D)
             \[3x-2y=3\sqrt{2}\] done clear
  View Solution play_arrow

• question_answer39)
  The curve given by \[x+y={{e}^{xy}}\] has a tangent parallel to the y-axis at the point                                     [AMU  2005]
  A)
             (0, 1) done clear
  B)
             (1, 0) done clear
  C)
                  (1, 1) done clear
  D)
                  (?1, ?1) done clear
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